Problem: The lifespans of sloths in a particular zoo are normally distributed. The average sloth lives $19.2$ years; the standard deviation is $3.5$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a sloth living longer than $29.7$ years.
Solution: $19.2$ $15.7$ $22.7$ $12.2$ $26.2$ $8.7$ $29.7$ $99.7\%$ $0.15\%$ $0.15\%$ We know the lifespans are normally distributed with an average lifespan of $19.2$ years. We know the standard deviation is $3.5$ years, so one standard deviation below the mean is $15.7$ years and one standard deviation above the mean is $22.7$ years. Two standard deviations below the mean is $12.2$ years and two standard deviations above the mean is $26.2$ years. Three standard deviations below the mean is $8.7$ years and three standard deviations above the mean is $29.7$ years. We are interested in the probability of a sloth living longer than $29.7$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the sloths will have lifespans within 3 standard deviations of the average lifespan. The remaining $0.3\%$ of the sloths will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({0.15\%})$ will live less than $8.7$ years and the other half $({0.15\%})$ will live longer than $29.7$ years. The probability of a particular sloth living longer than $29.7$ years is ${0.15\%}$.